Function Classes in the Functions and Modeling Pathway

Knowing key characteristics of the various function classes (e.g. linear, exponential, rational, polynomial) provides opportunities for students to expand their understandings of the ways in which two quantities can change together (and the associated patterns of change that might emerge). Understanding function classes in terms of these key characteristics can also expand students’ understanding of functions, facilitating a shift from thinking of a function as a procedure (in which each input is ‘plugged in’ to a particular formula to produce an ‘output’) to a broader representation of an entire relationship between two quantities. This has a number of advantages, one of which is that it supports thinking about constructions involving multiple functions (e.g. function composition and combinations of functions). Another benefit of viewing a function as a unified process is that it supports students’ abilities to compare and contrast the behavior of two functions against one another (in a way that is usually not possible if a student has only a computational input-output view of function). Such comparisons are integral to the Functions and Modeling course because the classification of a particular function as similar to particular class of functions provides a tool to analyze, model, and describe the behavior of a variety of real-world situations.

Faculty at the MIP Functions and Modeling Initiation Workshop in June, 2019 identified the following aspects of Function Classes as critical for success in the Oklahoma Functions and Modeling Math Pathway.

Covariational Reasoning: Activities should emphasize quantitative and covariational characterizations of each function class. As a function is a relationship between quantities, it is propitious for students to characterize types of functions by the covariational patterns that underpin them. For example, exponential functions can be characterized covariationally in several ways, including, for fixed, uniform changes in the input quantity, (1) as functions admit a constant percentage change (alternatively, growth factor) for uniform changes in input, and (2) as functions for which the change in instantaneous rate of change (alternatively, average rate of change) is proportional to the function value. Both of these characterizations support students’ ability to reason about how the quantities change together. There are several conceptual analyses in the literature that outline productive (quantitative, covariational) understandings for linear functions (e.g. Musgrave & Carlson, 2016; Thompson & Thompson, 1994; Thompson, 2008) and exponential functions (Ellis et al., 2012; O’Bryan, 2018; Thompson, 2008) that could be useful when identifying worthwhile targets of instruction.

Modeling with Rates: Activities should provide opportunities for students to interpret rate of change information from real-world scenarios – see Carlson et al.’s (2002) framework for examples of reasoning covariationally. Linear functions can be characterized as functions with a constant rate of change (in which the change in output is proportional to the change in input) as opposed to focusing on perceptual features (e.g. that the graph is a line). Such an image of constant rate could be used to interpolate/extrapolate unknown function values or estimate growth rates of other functions as if the function were linear. The limiting value and inflection point of a logistic function can be discussed in terms of what their respective rates of change mean within that particular situation (e.g. the inflection point occurs when the average rate of change is maximized, and the limiting value occurs when the average rate of change tends to 0, both of which underscore important information about population growth).

Multiple Representations: Activities should help students explore key characteristics of each function class (linear, exponential, polynomial, logarithmic, and rational) through the lens of different function representations (e.g. see Oehrtman, Carlson, & Thompson, 2008). For example, reflecting on a variety of
exponential functions across multiple representations can lead a student to recognize that all exponential functions can be expressed in the same algebraic form and the components of this formula correspond to important attributes of the problem situation, that the graphs of exponential functions have constant concavity (increasing/decreasing average rate of change), that the changes in the output quantity are proportional to the function value in a table, and that exponential functions can be interpreted through the lens of percentage change and/or growth factors. A student with each of these four understandings will be well-positioned to model exponential relationships, whereas a student with a minority of them will likely encounter difficulty.

Technology: Activities should use technology as a tool to assist students in developing the reasoning abilities outlined above. Technology (e.g. graphing calculators) can efficiently generate additional function representations and enable students to reflect on the common features of various representations more easily than they would otherwise. Generation of these additional representations is only helpful to students if they have quantitative and covariational understandings of them. Otherwise, the features that the students reflect on and abstract across representations might be superficial and lack quantitative meaning (e.g. Thompson, 2013). Instructional designers may consider emphasizing quantitative and covariational understandings of the various function representations early in, and throughout, their modules so that students notice and attend to aspects of quantitative relationships that will enable productive real-world interpretations.

Addressing Components of Inquiry

Participants of the MIP Workshop on Functions and Modeling suggested the following ways modules about Function Classes could address the three MIP components of mathematical inquiry:

Active Learning: Students should be engaged in tasks that go beyond treating functions as equations and that provide opportunities for them to create and interpret function models to solve novel problems. Tasks should invoke function classes and notation in ways responsive to that problem-solving activity.

Meaningful Applications: Activities could emphasize modeling and interpretation to reinforce functions as a tool to describe the world. The coordination of two quantities and univalence built into the mathematical structure of functions can gain compelling meaning from natural relationships and constraints between quantities in real world situations. One may ask students to contrast the domain and range of functions based on the problem context. Students should also identify and interpret key parameters in each function class in terms of the context in which it is being applied and in its various mathematical representations.

Academic Success Skills: When improperly motivated, introduction of functions can seem arbitrary and unnecessarily complicated, raising a barrier to many students. Modules should help students become confident in their use of functions as a foundation of the language of mathematics and science.

References

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 352-378.

Ellis, A. B., Ozgur, Z., Kulow, T., Williams, C., & Amidon, J. (2012). Quantifying exponential growth: The case of the jactus. Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context, 2, 93-112.

Musgrave, S., & Carlson, M. (2016). Transforming graduate students’ meanings for average rate of change. In Proceedings of the 19th meeting of the MAA special interest group on research in undergraduate mathematics education. Pittsburgh, PA.

O’Bryan, A. E. (2018). Exponential Growth and Online Learning Environments: Designing for and studying the development of student meanings in online courses. Arizona State University.

Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. Making the connection: Research and teaching in undergraduate mathematics education, 27, 42.

Thompson, P. W. (2013). In the absence of meaning…. In Vital directions for mathematics education research (pp. 57-93). Springer, New York, NY.

Thompson, P. W., & Thompson, A. G. (1994). Talking about rates conceptually. Journal for Research in Mathematics Education, 25(3), 279-303.

Additional Resources

Carlson, M., Oehrtman, M., & Moore, K. (2010). Precalculus: Pathways to calculus: A problem solving approach. Rational Reasoning.

Crauder, B., Evans, B., & Noell, A. (2013). Functions and change: A modeling approach to college algebra. Cengage Publishing.